This paper studies some properties of stochastic dominance (SD) for risk-averse and risk-seeking investors, especially for the third order SD (TSD). We call the former ascending stochastic dominance (ASD) and the latter descending stochastic dominance(DSD). We first discuss the basic property of ASD and DSD linking the ASD and DSD of the first three orders to expected-utility maximization for risk-averse and risk-seeking investors. Thereafter, we prove that a hierarchy exists in both ASD and DSD relationships and that the higher orders of ASD and DSD cannot be replaced by the lower orders of ASD and DSD. Furthermore, we study conditions in which third order ASD preferences will be 'the opposite of' or 'the same as' their counterpart third order DSD preferences. In addition, we construct examples to illustrate all the properties developed in this paper. The theory developed in this paper provides investors with tools to identify first, second, and third order ASD and DSD prospects and thus they could make wiser choices on their investment decision.